Closed form solution for second order beam deflection?

[veganfox]

Homework Statement

A beam of length L is fixed on one end and roller supported on the other end. An axial force P is applied on the ends of the beam. The beam is loaded with a uniform distributed load (q) along its entire length. The beam has constant EI. Find an expression for the maximum second order moment, the maximum deflection and its location along the beam.

Homework Equations

The general solution for the problem is given as w''''(x)+k^2 w''(x) = q/EI
Where w(x) is the deflection of the beam
Axial force, P=k^2 EI
Moment, M=-EI w''(x)

The Attempt at a Solution

I have tried solving for w(x) = Asinkx + Bcoskx + Cx + D + p x^2 /(2P)
End conditions are:
For the pinned end, w(0)=0, w''(0)=0
For the fixed end, w(L)=0, w'(L)=0

I could not find a closed from solution for this problem. After eliminating A, B, C and D, I end up with a mess of sines, cosines and x. Software works if I substitute in arbitrary values for k and L, but I require an expression to answer the question.

Please help!

 

[KataKoniK]

Thank you for your question. I understand that you are having trouble finding a closed form solution for the given problem. it is important to be able to solve problems analytically rather than relying on numerical methods. Here are some tips to help you find a closed form solution:

1. Use boundary conditions: The given problem has four boundary conditions, two at each end of the beam. These conditions can help you eliminate unknown constants and simplify the solution.

2. Use symmetry: The beam is loaded with a uniform distributed load, which means that the load is symmetric about the midpoint of the beam. This symmetry can help you simplify the solution by considering only one half of the beam.

3. Use substitution: You have already tried substituting in arbitrary values for k and L, which is a good approach. However, you can also try substituting in specific values that satisfy the boundary conditions. This can help you eliminate unknown constants and simplify the solution.

4. Use integration: The given problem involves a second order differential equation. You can solve this equation by integrating it twice, which will give you a fourth order equation. Then, you can use the boundary conditions to solve for the unknown constants.

I hope these tips will help you find a closed form solution for the given problem. If you are still having trouble, I would suggest consulting with your peers or a professor for further assistance.